Multi-Energy CT Phase Retrieval

• Multi-Energy CT phase retrieval is a technique that recovers energy-dependent phase-shifting and attenuation properties, enabling visualization of weakly absorbing structures.
• It employs basis-material decomposition and both projection- and post-reconstruction methods to integrate multi-energy constraints into a unified inversion framework.
• Recent advancements favor optimization-based, one-step 3D inversion with spectral modeling for improved quantitative accuracy and image quality.

Searching arXiv for recent and foundational papers on multi-energy CT phase retrieval, propagation-based phase-contrast CT, and related reconstruction frameworks. Multi-Energy CT phase retrieval denotes the recovery of energy-dependent phase-shifting and absorbing object properties from computed tomography data acquired at multiple X-ray energies. In the phase-contrast setting, the object is described by an energy-dependent complex refractive index, and the inversion target may be the refractive index decrement, the attenuation-related term, or basis-material densities from which both are derived. Across the literature, the topic spans propagation-based X-ray phase-contrast CT, dual- and triple-energy material decomposition, post-reconstruction and projection-domain phase retrieval, and optimization-based 3D inversion. A central unifying theme is that multi-energy measurements provide complementary constraints on the energy dependence of attenuation and phase, while phase-contrast physics provides sensitivity to weakly absorbing structures that are difficult to visualize with absorption-only CT (Jadick et al., 17 Aug 2025).

1. Conceptual scope and physical model

In conventional X-ray CT, the object is modeled through the linear attenuation coefficient, whereas phase-contrast imaging exploits the wave nature of X-rays. In the propagation-based formulation, the object is described by the energy-dependent complex refractive index

$$n_E(\mathbf{r}) = 1 - \delta_E(\mathbf{r}) - i \beta_E(\mathbf{r}),$$

where $\delta_E(\mathbf{r})$ is the refractive index decrement and $\beta_E(\mathbf{r})$ is the absorption term; the attenuation coefficient is related by

$\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$

with $k_E$ the wavenumber (Jadick et al., 17 Aug 2025). Phase retrieval in this context means recovering $\delta_E(\mathbf{r})$, and often implicitly $\mu_E(\mathbf{r})$, from intensity measurements after free-space propagation.

Under the projection approximation used in propagation-based X-ray phase-contrast CT, the exit wave is written as

$$\psi^{(\mathrm{obj})}_E(\mathbf{r}_\perp) = \psi^{(\mathrm{in})}_E(\mathbf{r}_\perp) \exp\left[ - \int \mu_E(\mathbf{r})\, dz - i k_E \int \delta_E(\mathbf{r})\, dz \right].$$

After propagation over a distance $R$, the field is convolved with the Fresnel kernel

$$h_{R,E}(\mathbf{r}_\perp) = \exp\left[ - \frac{i k_E}{2 R} (x^2 + y^2) \right],$$

and the detector measures the propagated intensity magnitude (Jadick et al., 17 Aug 2025). This formulation makes explicit that measured intensity depends jointly on attenuation and phase.

The multi-energy component arises because $\delta_E(\mathbf{r})$0 and $\delta_E(\mathbf{r})$1 are energy dependent. In the material-basis viewpoint, attenuation can be written as a linear combination of basis functions,

$\delta_E(\mathbf{r})$2

with a two-term form for classical diagnostic materials and a three-term extension when a K-edge material is present (Zhao et al., 2018). A closely analogous decomposition can be posed for phase-shifting properties,

$\delta_E(\mathbf{r})$3

so that multi-energy phase retrieval becomes a parameter-retrieval problem over basis-material densities or related coefficients. This suggests that the mathematical structure developed for dual- and triple-energy attenuation imaging is directly relevant to phase retrieval, even when the original formulation is attenuation only (Zhao et al., 2018).

2. Basis-material decomposition and the transition from attenuation to phase

A major line of work in multi-energy CT concerns quantitative material decomposition. In a unified dual- and triple-energy framework, the measured intensity at a detector pixel is modeled by a polychromatic Beer–Lambert law with energy-integrating detector response, and the predicted noise-free log-projection is

$\delta_E(\mathbf{r})$4

where $\delta_E(\mathbf{r})$5 are line integrals of basis-material densities, $\delta_E(\mathbf{r})$6 for dual-energy CT and $\delta_E(\mathbf{r})$7 for triple-energy CT (Zhao et al., 2018). The same paper formulates material decomposition as a constrained nonlinear least-squares problem with non-negativity,

$\delta_E(\mathbf{r})$8

solved ray by ray and followed by filtered backprojection to reconstruct each basis-material image (Zhao et al., 2018).

That framework is attenuation based, but it is structurally important for multi-energy phase retrieval. The same paper explicitly notes that basis-material parametrization, polychromatic integration, and nonlinear least-squares inversion are generic and can be extended so that $\delta_E(\mathbf{r})$9 and $\beta_E(\mathbf{r})$0 are both decomposed in shared material bases, with attenuation measurements constraining one set of coefficients and phase-sensitive measurements constraining another (Zhao et al., 2018). A plausible implication is that multi-energy CT phase retrieval can be cast as a joint inverse problem over material densities rather than as a separate recovery of phase and attenuation images.

This basis-material perspective is especially important when contrast agents or K-edge materials are present. The dual-/triple-energy framework reports that triple-energy CT separates gadodiamide and iodine and that virtual monochromatic images synthesized from basis-material images show superior image quality compared to polychromatic kV CT images (Zhao et al., 2018). This suggests that the same strategy could be used in phase-contrast settings to distinguish materials whose attenuation and dispersion signatures differ with energy.

3. One-step and optimization-based formulations

A decisive development in the field is the move from sequential pipelines to one-step or natively 3D optimization. In “Optimization-based phase retrieval for material decomposition with multi-energy computed tomography” (Jadick et al., 17 Aug 2025), the unknowns are the 3D basis material density volumes

$\beta_E(\mathbf{r})$1

with $\beta_E(\mathbf{r})$2 in the simulations, corresponding to soft tissue and bone. These densities determine $\beta_E(\mathbf{r})$3 and $\beta_E(\mathbf{r})$4 via known spectral properties, so the reconstruction does not separately estimate phase maps and attenuation maps; instead, both are implicit in the material densities (Jadick et al., 17 Aug 2025).

The optimization problem is formulated as

$\beta_E(\mathbf{r})$5

where $\beta_E(\mathbf{r})$6 is the measured multi-energy propagation-based phase-contrast CT data, $\beta_E(\mathbf{r})$7 is the forward model combining projection approximation and Fresnel propagation, and the regularization combines sparsity, total variation, and non-negativity (Jadick et al., 17 Aug 2025). The paper uses automatic differentiation through the JAX-based wave-optics library chromatix and optimizes with Adam at learning rate $\beta_E(\mathbf{r})$8 (Jadick et al., 17 Aug 2025).

The importance of this formulation lies in its integration of phase retrieval, tomography, and material decomposition into a single inverse problem. Earlier analytical multi-energy propagation-based methods are described in the same paper as projection based, relying on transport-of-intensity equation linearization and separate tomographic reconstruction (Jadick et al., 17 Aug 2025). By contrast, the optimization-based method uses the full propagation-based phase-contrast Fresnel model and solves directly in 3D, thereby avoiding many analytical approximations (Jadick et al., 17 Aug 2025).

A related direction appears in the dual-energy propagation-based phase-contrast CT literature, where one-step methods are explicitly contrasted with “pre-reconstruction and post-reconstruction (two-step)” approaches, and the one-step iterative method is described as performing phase retrieval, reconstruction, and material decomposition directly from the intensity data in different energies (Liao et al., 2023). Although the detailed content for that paper is not available in the supplied record, the stated distinction between one-step and two-step methods aligns with the optimization-based 3D trend (Liao et al., 2023).

4. Post-reconstruction versus projection-domain phase retrieval

Another major axis of the subject is whether phase retrieval is performed on 2D projections before CT or on the reconstructed 3D volume afterward. In the contrast-transfer-function formalism for propagation-based phase-contrast tomography, the object transmission for angle $\beta_E(\mathbf{r})$9 is

$\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$0

with $\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$1 and $\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$2 under the projection approximation (Thompson et al., 2022). Linearized CTF yields a Fourier-domain relation between measured contrast and projected absorption and phase, and the method can be extended from 2D projection-space retrieval to a 3D post-reconstruction formulation (Thompson et al., 2022).

The key 3D CTF relation is

$\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$3

which is the volumetric analogue of the 2D CTF equation (Thompson et al., 2022). The paper shows that post-reconstruction 3D CTF retrieval produces results equivalent to conventional pre-reconstruction 2D CTF retrieval, while allowing the operation to be highly localized to isolated objects of interest (Thompson et al., 2022).

For monomorphous objects, the CTF-Hom case uses a single propagation distance and a constant ratio $\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$4, leading to a one-distance 3D inversion formula (Thompson et al., 2022). The same work also extends the formalism to partially coherent illumination and strongly absorbing samples through modified 3D propagators (Thompson et al., 2022). This is important because it moves 3D phase retrieval beyond ideal weak-object assumptions and makes post-reconstruction strategies more relevant to realistic CT systems.

The post-reconstruction viewpoint intersects naturally with multi-material and potentially multi-energy settings. The paper states that for objects containing several distinct components localized to separate 3D regions, one may perform the expensive CT reconstruction once and then repeatedly apply local phase retrieval with different local $\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$5 values in each region (Thompson et al., 2022). This suggests that, in a spectral or multi-energy setting, one could similarly use energy-dependent $\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$6 or basis-material relations to apply local volumetric retrieval conditioned on material class.

5. Robust and efficient solvers for practical regimes

Practical phase retrieval is shaped by robustness to strong attenuation, heterogeneous materials, and computational cost. One notable non-iterative approach is “3D Masked Phase Retrieval (3DMPR)” for propagation-based CT near highly attenuating objects (Pollock et al., 2023). The method begins from the observation that Paganin-type single-material retrieval uses a Lorentzian low-pass filter and that, in multi-material samples, tuning the filter for the boundary with strongest phase contrast can over-blur other interfaces (Pollock et al., 2023). The remedy is to mask highly attenuating regions in the CT volume, replace them by a uniform low-$\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$7 value, apply strong single-material 3D phase retrieval to the masked volume, and then reinsert the high-$\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$8 material from an interface-tuned reconstruction (Pollock et al., 2023).

Quantitatively, the paper reports a 6.9-fold improvement in the signal-to-noise ratio of brain tissue compared to the standard phase retrieval procedure for rabbit kitten brain data acquired with 24 keV synchrotron radiation and a 5 m propagation distance, without over-smoothing the images (Pollock et al., 2023). For an aluminium-water phantom, the method provided a 4.2-fold SNR boost while preserving the boundary resolution at $\mu_E(\mathbf{r}) = 2 k_E \beta_E(\mathbf{r}),$9, compared to $k_E$0 in conventional phase retrieval (Pollock et al., 2023). These results address a common misconception that strong phase retrieval and boundary preservation are necessarily incompatible; the method shows that the trade-off can be spatially managed by material-aware masking.

At the opposite end of the modeling spectrum, “Fast Eikonal Phase Retrieval for High-Throughput Beamlines” introduces a formulation that accelerates eikonal phase retrieval by more than two orders of magnitude while retaining controlled accuracy (Mirone et al., 30 Jan 2026). The method combines a local $k_E$1 closure for sub-pixel eikonal shifts with a non-local formulation for multi-pixel shifts based on an explicit eikonal ray mapping, mass-conserving bilinear redistribution on the detector grid, and corresponding adjoint interpolation back to the object grid (Mirone et al., 30 Jan 2026). The same framework supports polychromatic data through a compact spectral discretisation, allowing energy-dependent transport and inversion while keeping the iteration GPU/FFT efficient (Mirone et al., 30 Jan 2026).

This solver family is directly relevant to multi-energy CT phase retrieval because it explicitly treats energy dependence in both transport and inversion. The paper models polychromatic detector intensity as a sum over spectral components and applies energy-dependent Paganin-type inverse operators as FFT-diagonal preconditioners (Mirone et al., 30 Jan 2026). A plausible implication is that high-throughput multi-energy propagation-based CT can combine wave-optics-aware phase retrieval with spectral modeling at computational costs closer to standard Paganin filtering than to full Fresnel inversion.

6. Regularization, structural priors, and relation to spectral CT reconstruction

Multi-energy phase retrieval does not exist in isolation from broader multi-spectral CT reconstruction. One relevant contribution is the synergistic reconstruction framework based on directional total variation (Cueva et al., 2021). That work constructs a single “polyenergetic” image from a fused sinogram

$k_E$2

reconstructs a high-SNR structural prior $k_E$3 through TV-regularized inversion, and then uses $k_E$4 to define a directional total variation penalty for each individual energy channel (Cueva et al., 2021). The channel-wise problems take the form

$k_E$5

with $k_E$6 chosen as TV or directional TV (Cueva et al., 2021).

Although this paper does not address phase contrast explicitly, it is directly relevant as a regularization backbone. It shows that structural information shared across energies can be encoded in a polyenergetic prior and that directional total variation promotes alignment of gradients with that prior (Cueva et al., 2021). The authors note that such structural priors could be applied not only to attenuation images but also to phase maps, for example by defining a DTV-like regularizer for a phase image using the same edge directions extracted from the polyenergetic image (Cueva et al., 2021). This suggests that multi-energy CT phase retrieval can benefit from structural coupling across energy channels even when the forward physics is more complex than linear attenuation.

A separate but related line comes from deep-learning-based correction of current-integrating polychromatic CT data into monochromatic projections (Cong et al., 2017). That work learns a nonlinear transformation mapping path-wise values extracted from a conventional reconstruction to monochromatic projection values consistent with a linear Radon model, achieving projection correction with a relative error of less than 0.2% (Cong et al., 2017). The paper is attenuation focused, but it illustrates a general strategy: use multi-energy or dual-energy data to learn a nonlinear “physics correction” that transforms raw measurements into a form suitable for standard inversion. A plausible implication is that analogous networks could be trained to map multi-energy phase-contrast measurements to line integrals of $k_E$7, $k_E$8, or basis-material coefficients, using physics-based phase retrieval as supervision.

7. Quantitative evidence, limitations, and emerging directions

The most direct quantitative evidence for multi-energy CT phase retrieval presently comes from optimization-based propagation-based studies. In the 3D optimization framework for material decomposition with multi-energy CT, a simulation with a soft-tissue cylinder containing embedded bone spheres compared the optimization-based method against a TIE-based analytical method plus filtered backprojection (Jadick et al., 17 Aug 2025). For tissue, the optimization method achieved SSIM/RMSE values of $k_E$9 at $\delta_E(\mathbf{r})$0 mm, $\delta_E(\mathbf{r})$1 at $\delta_E(\mathbf{r})$2 mm, and $\delta_E(\mathbf{r})$3 at $\delta_E(\mathbf{r})$4 mm, while the TIE-based method yielded $\delta_E(\mathbf{r})$5, $\delta_E(\mathbf{r})$6, and $\delta_E(\mathbf{r})$7, respectively (Jadick et al., 17 Aug 2025). For bone, the optimization method achieved $\delta_E(\mathbf{r})$8, $\delta_E(\mathbf{r})$9, and $\mu_E(\mathbf{r})$0, compared with $\mu_E(\mathbf{r})$1, $\mu_E(\mathbf{r})$2, and $\mu_E(\mathbf{r})$3 for the analytical method (Jadick et al., 17 Aug 2025). The same study also shows that without phase contrast ($\mu_E(\mathbf{r})$4), internal voids in weakly absorbing tissue are not recovered, whereas with phase contrast ($\mu_E(\mathbf{r})$5) they become reconstructible (Jadick et al., 17 Aug 2025).

These results support two conclusions. First, multi-energy phase retrieval is not merely a spectral extension of attenuation imaging; phase contrast changes what is detectable, especially for weakly absorbing structures (Jadick et al., 17 Aug 2025). Second, projection-domain analytical methods based on transport-of-intensity linearization have limited validity regimes, especially for larger propagation distances and stronger phase gradients (Jadick et al., 17 Aug 2025).

At the same time, limitations are explicit across the literature. The optimization-based multi-energy PB-XPC work assumes the projection approximation, neglects scattering, uses monochromatic simulations at 24 keV and 34 keV, and adopts a simplified detector model with Lorentzian blur and Poisson noise (Jadick et al., 17 Aug 2025). The 3D CTF formalism relies on linearization assumptions, though it is extended to partial coherence and stronger absorption (Thompson et al., 2022). Eikonal approaches require near-field, single-valued ray mapping conditions and separate local and non-local solvers depending on the shift-to-pixel regime (Mirone et al., 30 Jan 2026). Material-aware masking methods require segmentation or thresholding of materials in CT space and can be affected by beam hardening or overlapping attenuation distributions (Pollock et al., 2023).

A persistent controversy concerns whether phase retrieval should be regarded primarily as a projection-domain preprocessing step or as part of a fully coupled inverse problem. The current literature indicates that both views remain active. Post-reconstruction CTF methods demonstrate equivalence to pre-reconstruction retrieval under stated assumptions while enabling locality and computational savings (Thompson et al., 2022). One-step optimization methods argue that decoupling phase retrieval, CT reconstruction, and material decomposition sacrifices accuracy because intensity data cannot feed back into the volume estimate during inversion (Jadick et al., 17 Aug 2025). This suggests that the choice of formulation depends strongly on the intended operating regime: fast local processing and weak-scattering settings favor CTF-type pipelines, while quantitative multi-material recovery in low-contrast regimes favors integrated optimization.

Overall, multi-energy CT phase retrieval has evolved from a conceptual combination of spectral CT and phase-contrast imaging into a set of concrete inversion strategies: basis-material decomposition rooted in energy dependence, CTF- and TIE-based retrieval in projection or volume domains, masking strategies for heterogeneous samples, and fully differentiable 3D optimization using Fresnel physics. The field’s current trajectory suggests tighter integration of spectral modeling, wave-optics forward operators, and structural regularization, with explicit attention to realistic scanner configurations, computational scalability, and quantitative material decomposition (Zhao et al., 2018, Cueva et al., 2021, Jadick et al., 17 Aug 2025).